Error Correction Codes (ECC) are used in a variety of applications, such as in various digital communication and data storage applications. Some ECC decoders apply a multistage process, which decodes ECC code words by calculating syndromes of the code words, using the syndromes to generate Error Locator Polynomials (ELPs) whose roots indicate the error locations in the code words, finding the ELP roots, and thus correcting the errors. Some ECC types that are commonly decoded using such a process comprise, for example, Bose-Chaudhuri-Hocquenghem (BCH) codes and Reed-Solomon (RS) codes.
Various schemes for generating ELPs from syndromes are known in the art. Some well-known schemes comprise, for example, the Berlekamp-Massey algorithm, the Euclidean algorithm and the Peterson Gorenstein Zierler algorithm. Examples of methods for determining ELPs are described, for example, by Lin and Costello in “Error Control Coding Fundamentals,” Prentice Hall, second edition, 2004, chapter 6, pages 209-215, and chapter 7, pages 241-255; and by Blahut in “Algebraic Codes for Data Transmission,” Cambridge University Press, 2004, chapter 6, pages 131-166, and chapter 7, pages 179-190 and 217-223, which are incorporated herein by reference.
A method for finding ELP roots is described by Chien in “Cyclic Decoding Procedure for the Bose-Chaudhuri-Hocquenghem Codes,” IEEE Transactions on Information Theory, vol. IT-10, October, 1964, pages 357-363, which is incorporated herein by reference. This method is commonly known as the “Chien search.” Chen and Parhi describe hardware configurations that perform parallel Chien search, in “Small Area Parallel Chien Search Architectures for Long BCH Codes,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, Vol. 12, No. 5, May 2004, pages 545-549, which is incorporated herein by reference.
U.S. Patent Application Publication 2010/0332955, whose disclosure is incorporated herein by reference, describes a method for decoding an ECC. The method includes accepting coefficients, including at least first and second coefficients, of an ELP that is defined over a vector space and has at least one root that is indicative of a location of an error in a set of bits, which represent data that has been encoded with the ECC. The first coefficient is represented using a first basis of the vector space, and the second coefficient is represented using a second basis of the vector space, different from the first basis. Using processing circuitry, the root of the ELP is identified by applying algebraic operations to the coefficients, such that the algebraic operations are applied to the first coefficient using the first basis, and to the second coefficient using the second basis. The error is corrected responsively to the identified root of the ELP.